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Op
ti
m
a
l
p
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we
r
fl
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(OPF
)
is
a
c
o
m
p
lex
,
n
o
n
-
li
n
e
a
r
o
p
ti
m
iza
ti
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ro
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lem
fo
c
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se
d
o
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d
e
term
in
in
g
th
e
ste
a
d
y
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ra
m
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ters
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s
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e
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o
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o
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ic
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n
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se
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o
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e
ra
ti
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n
.
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h
e
c
h
a
ll
e
n
g
e
in
ten
sif
ies
d
u
e
t
o
n
u
m
e
ro
u
s s
y
ste
m
c
o
n
stra
in
ts t
h
a
t
m
u
st b
e
sa
ti
sfie
d
sim
u
lt
a
n
e
o
u
sly
.
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o
u
g
h
v
a
rio
u
s
e
v
o
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ti
o
n
a
r
y
a
lg
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m
s
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a
v
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a
p
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li
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d
t
o
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in
re
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e
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t
d
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a
d
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th
e
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m
s
o
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e
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u
se
u
n
c
o
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stra
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e
d
se
a
rc
h
stra
teg
ies
.
A
c
o
m
m
o
n
a
p
p
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c
h
t
o
h
a
n
d
le
c
o
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stra
i
n
t
v
i
o
latio
n
s
is
th
e
sta
ti
c
p
e
n
a
lt
y
fu
n
c
ti
o
n
,
w
h
ich
p
e
n
a
li
z
e
s
in
fe
a
sib
le
so
lu
t
io
n
s.
Ho
we
v
e
r,
se
lec
ti
n
g
su
it
a
b
le
p
e
n
a
lt
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c
o
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ts
t
y
p
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ll
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ti
m
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n
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m
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l
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r,
a
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ti
n
g
o
v
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ra
ll
p
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rfo
rm
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n
c
e
.
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is
stu
d
y
e
x
p
l
o
re
s
th
e
i
n
te
g
ra
ti
o
n
o
f
a
d
v
a
n
c
e
d
c
o
n
stra
in
t
h
a
n
d
li
n
g
(CH)
tec
h
n
iq
u
e
s
with
i
n
t
h
e
d
iffere
n
ti
a
l
e
v
o
lu
ti
o
n
(DE)
fra
m
e
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e
n
h
a
n
c
e
t
h
e
p
e
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rm
a
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c
e
o
f
o
p
ti
m
a
l
p
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flo
w
(OPF
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so
l
u
ti
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n
s.
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n
p
a
rti
c
u
lar,
it
l
o
o
k
s
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t
t
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re
e
a
p
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e
s:
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h
y
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rid
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n
se
m
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le
o
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CH
tec
h
n
iq
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(ECHT),
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se
lf
-
a
d
a
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ti
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e
th
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n
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p
e
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o
f
v
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le so
l
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ti
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s (S
F
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.
Th
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E
3
0
-
b
u
s a
n
d
IEE
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-
57
b
u
s
b
e
n
c
h
m
a
rk
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ste
m
s
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re
u
se
d
to
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iq
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issio
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s
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n
d
g
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e
ra
ti
o
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c
o
sts,
c
u
tt
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s,
a
n
d
e
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h
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n
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to
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s.
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e
sim
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latio
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tco
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e
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in
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ica
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t
th
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se
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CH
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DE
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p
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s
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li
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ro
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st
a
n
d
c
o
m
p
e
ti
t
iv
e
o
p
ti
m
iza
ti
o
n
re
su
lt
s,
d
e
m
o
n
stra
ti
n
g
imp
ro
v
e
d
c
o
n
stra
i
n
t
h
a
n
d
li
n
g
c
a
p
a
b
il
i
ti
e
s
wh
e
n
c
o
m
p
a
re
d
to
c
o
n
tem
p
o
ra
ry
m
e
th
o
d
s in
th
e
l
it
e
r
a
tu
re
.
K
ey
w
o
r
d
s
:
C
o
n
s
tr
ain
t h
an
d
lin
g
Dif
f
er
en
tial e
v
o
lu
tio
n
E
v
o
lu
tio
n
a
r
y
alg
o
r
ith
m
s
Mu
lti
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o
b
jectiv
e
o
p
tim
izatio
n
Op
tim
al
p
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wer
f
lo
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T
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is i
s
a
n
o
p
e
n
a
c
c
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ss
a
rticle
u
n
d
e
r th
e
CC B
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li
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se
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C
o
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r
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s
p
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A
uth
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r
:
Sid
d
h
ar
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s
in
g
h
K.
C
h
au
h
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Dep
ar
tm
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t o
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Nir
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Un
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d
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ail:
s
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s
in
g
h
.
c
h
au
h
a
n
@
n
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m
au
n
i.a
c.
in
1.
I
NT
RO
D
UCT
I
O
N
T
h
e
o
p
tim
al
p
o
wer
f
lo
w
(
OPF
)
p
r
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b
lem
b
ee
n
a
f
o
ca
l
p
o
in
t
o
f
in
v
esti
g
ate
in
p
o
wer
s
y
s
tem
s
f
o
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m
o
r
e
th
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f
if
ty
y
ea
r
s
d
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e
to
its
in
h
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en
t
co
m
p
lex
ity
an
d
r
ea
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ld
im
p
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ce
.
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o
r
e,
OPF
in
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lv
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m
in
in
g
t
h
e
h
ig
h
est
o
p
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atin
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co
n
d
itio
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o
f
a
p
o
wer
g
r
id
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y
m
i
n
im
izin
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b
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s
u
ch
as
f
u
el
co
s
ts
,
em
is
s
io
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s
,
ac
tiv
e
p
o
wer
l
o
s
s
es,
an
d
v
o
ltag
e
d
ev
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n
s
,
wh
il
e
s
im
u
ltan
eo
u
s
ly
a
d
h
er
in
g
to
v
ar
io
u
s
en
g
in
ee
r
in
g
an
d
o
p
e
r
atio
n
al
co
n
s
tr
ain
ts
.
T
h
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in
clu
d
e
g
e
n
er
ato
r
o
u
tp
u
t
lim
its
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v
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ltag
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b
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n
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s
,
th
er
m
al
ca
p
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ities
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f
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an
s
m
is
s
io
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lin
es,
an
d
p
o
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ir
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ts
.
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tim
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o
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p
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an
d
r
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p
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m
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[
1
]
.
I
n
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d
itio
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,
r
ea
ctiv
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p
o
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s
u
p
p
o
r
t,
o
f
te
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s
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k
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u
latio
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esp
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ctiv
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lo
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i
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co
n
d
itio
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s
.
T
r
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al
m
ath
em
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al
o
p
ti
m
izatio
n
tech
n
iq
u
es
s
tr
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g
g
led
with
s
o
lv
in
g
OPF
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
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2
2
5
2
-
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7
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2
I
n
t J Ap
p
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n
g
,
Vo
l.
1
5
,
No
.
2
,
J
u
n
e
20
2
6
:
663
-
673
664
p
r
o
b
lem
s
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1
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T
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ce
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tio
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ased
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eth
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d
s
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v
e
s
h
o
wn
g
r
ea
ter
s
u
cc
ess
in
escap
in
g
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l
tr
ap
s
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d
id
e
n
tify
i
n
g
g
lo
b
ally
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o
m
p
etitiv
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lu
ti
o
n
s
.
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h
e
liter
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r
e
r
ef
lects
a
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ce
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o
b
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s
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co
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s
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h
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[
2
]
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ad
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with
v
alv
e
-
p
o
in
t
ef
f
ec
ts
,
an
d
im
p
r
o
v
e
d
co
llid
in
g
b
o
d
ies
.
Fu
r
th
er
in
n
o
v
atio
n
s
s
u
ch
a
s
th
e
m
o
th
s
war
m
alg
o
r
ith
m
[
3
]
,
ch
ao
tic
ar
tific
ial
b
e
e
co
lo
n
y
,
g
r
av
itatio
n
al
s
ea
r
ch
al
g
o
r
ith
m
,
an
d
a
d
ap
tiv
e
b
i
o
g
eo
g
r
ap
h
y
-
b
ased
o
p
tim
izatio
n
[
3
]
,
r
ef
lec
t
th
e
d
iv
er
s
ity
an
d
cr
ea
tiv
ity
in
tack
lin
g
OPF
ch
allen
g
es.
Oth
er
h
y
b
r
id
o
r
n
at
u
r
e
-
in
s
p
ir
ed
m
eth
o
d
s
,
in
clu
d
i
n
g
f
u
zz
y
h
ar
m
o
n
y
s
ea
r
ch
[
4
]
,
teac
h
in
g
-
lear
n
i
n
g
-
b
ased
with
L
év
y
m
u
tatio
n
[
4
]
,
an
d
Kr
ill
Her
d
a
lg
o
r
ith
m
s
u
n
d
er
s
co
r
e
th
e
wid
esp
r
ea
d
ac
ce
p
tan
ce
.
Am
o
n
g
th
ese,
d
if
f
er
e
n
tial
ev
o
lu
ti
o
n
(
DE
)
an
d
its
m
o
d
if
ied
f
o
r
m
s
h
av
e
g
ain
e
d
co
n
s
id
er
ab
le
atten
tio
n
,
p
ar
ti
cu
lar
ly
in
r
ea
l
-
co
d
ed
s
in
g
l
e
-
o
b
jectiv
e
p
r
o
b
lem
s
,
o
win
g
to
th
eir
s
tr
o
n
g
p
er
f
o
r
m
an
ce
in
g
lo
b
al
o
p
tim
iz
atio
n
b
e
n
ch
m
ar
k
s
[
5
]
.
I
n
s
p
ir
e
d
b
y
th
ese
s
u
cc
ess
es,
th
is
wo
r
k
ad
o
p
ts
DE
as
th
e
co
r
e
o
p
tim
izatio
n
tech
n
iq
u
e
f
o
r
s
o
lv
in
g
v
ar
i
o
u
s
OPF
s
ce
n
ar
io
s
.
Desp
ite
ad
v
an
ce
m
e
n
ts
in
s
ea
r
ch
tech
n
i
q
u
es,
o
n
e
m
ajo
r
asp
ec
t
th
at
r
em
ain
s
cr
u
cial
is
co
n
s
tr
ain
t
h
an
d
lin
g
(
C
H)
.
E
f
f
ec
tiv
e
C
H
s
tr
ateg
ies
ar
e
ess
en
tial
to
en
s
u
r
e
th
at
ca
n
d
id
ate
s
o
lu
tio
n
s
ar
e
n
o
t
o
n
ly
o
p
tim
al
b
u
t
also
f
ea
s
ib
le
u
n
d
er
all
s
y
s
tem
co
n
s
tr
ain
ts
.
Ma
n
y
p
ast
ap
p
r
o
ac
h
es
h
av
e
d
e
p
en
d
e
d
h
ea
v
ily
o
n
s
tatic
p
en
alty
f
u
n
ct
io
n
s
o
r
d
is
ca
r
d
e
d
in
f
ea
s
ib
le
s
o
lu
tio
n
s
alto
g
eth
er
.
W
h
ile
ea
s
y
to
im
p
lem
en
t,
th
ese
m
eth
o
d
s
co
m
e
with
lim
it
atio
n
s
:
im
p
r
o
p
e
r
ly
tu
n
ed
p
en
alty
p
ar
am
eter
s
ca
n
eith
er
p
er
m
it
ex
ce
s
s
iv
e
ex
p
lo
r
atio
n
o
f
in
f
ea
s
ib
le
s
p
ac
e
o
r
o
v
er
-
r
estrict
th
e
s
ea
r
ch
,
ca
u
s
in
g
ea
r
ly
s
tag
n
atio
n
.
On
th
e
o
t
h
er
h
an
d
,
elim
in
ati
n
g
in
f
e
asib
le
in
d
iv
id
u
als
f
r
o
m
th
e
p
o
p
u
latio
n
r
e
d
u
ce
s
d
i
v
er
s
ity
an
d
m
ak
es
it
h
ar
d
er
to
f
in
d
b
o
u
n
d
a
r
ies
b
et
wee
n
f
ea
s
ib
le
an
d
in
f
ea
s
ib
le
r
eg
io
n
s
[
5
]
.
T
o
o
v
e
r
co
m
e
th
e
s
e
ch
allen
g
es,
th
is
s
tu
d
y
im
p
lem
e
n
ts
an
d
e
v
alu
a
tes
th
r
ee
m
o
d
e
r
n
C
H
s
tr
ateg
i
es:
s
u
p
er
io
r
ity
o
f
f
ea
s
ib
le
s
o
l
u
tio
n
s
(
SF
)
,
wh
ich
p
r
io
r
itizes
f
ea
s
ib
le
s
o
lu
tio
n
s
d
u
r
in
g
s
elec
tio
n
,
s
elf
-
ad
a
p
tiv
e
p
en
alty
(
SP
)
,
wh
ich
d
y
n
am
ica
lly
ad
ju
s
ts
p
en
alty
weig
h
ts
d
u
r
in
g
th
e
r
u
n
,
an
d
en
s
em
b
le
co
n
s
tr
ain
t
-
h
an
d
lin
g
tech
n
iq
u
e
(
E
C
HT
)
,
a
h
y
b
r
id
ap
p
r
o
ac
h
th
at
co
m
b
in
es
SF
an
d
SP
in
a
d
y
n
am
ic
an
d
ad
ap
tiv
e
f
r
am
ew
o
r
k
.
T
h
e
E
C
HT
m
eth
o
d
is
in
tr
o
d
u
c
ed
with
a
p
r
ac
tical
p
er
s
p
ec
tiv
e:
s
in
ce
n
o
s
in
g
le
C
H
s
tr
ateg
y
co
n
s
is
ten
tly
ex
ce
ls
in
all
s
ce
n
a
r
io
s
,
co
m
b
in
in
g
m
u
ltip
le
ap
p
r
o
ac
h
es
ca
n
o
f
f
e
r
ad
a
p
tiv
e
r
o
b
u
s
tn
ess
,
r
ed
u
cin
g
th
e
n
ee
d
f
o
r
m
an
u
a
l
tu
n
in
g
o
r
al
g
o
r
ith
m
s
elec
tio
n
b
y
th
e
u
s
er
.
T
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
s
ar
e
test
ed
ac
r
o
s
s
a
v
ar
iety
o
f
o
p
tim
iz
atio
n
o
b
jectiv
es
,
in
clu
d
in
g
s
i
n
g
le
-
o
b
jectiv
e
ca
s
es
(
f
u
el
co
s
t,
em
is
s
io
n
s
,
lo
s
s
e
s
,
v
o
ltag
e
s
tab
ilit
y
)
an
d
m
u
lti
-
o
b
jectiv
e
ca
s
es
u
s
in
g
a
weig
h
t
ed
s
u
m
ap
p
r
o
ac
h
.
Simu
latio
n
o
u
tco
m
es
ar
e
th
o
r
o
u
g
h
ly
an
aly
ze
d
an
d
s
tatis
tica
lly
b
en
ch
m
a
r
k
ed
ag
ain
s
t
r
ec
e
n
tly
p
u
b
lis
h
ed
OPF
r
esu
lts
,
with
p
ar
ticu
lar
atten
tio
n
g
iv
en
to
co
n
s
tr
ain
t satis
f
ac
ti
o
n
an
d
s
o
lu
tio
n
r
eliab
ilit
y
[
5
]
.
2.
O
P
T
I
M
AL
P
O
W
E
R
F
L
O
W
M
AT
H
E
M
AT
I
CA
L
F
O
RM
UL
A
T
I
O
N
T
h
e
OPF
p
r
o
b
lem
is
a
co
m
p
l
ex
task
ch
ar
ac
ter
ized
b
y
its
n
o
n
-
lin
ea
r
a
n
d
n
o
n
-
c
o
n
v
e
x
n
at
u
r
e.
I
t
aim
s
to
o
p
tim
ize
s
p
ec
if
ic
p
er
f
o
r
m
an
ce
o
b
jectiv
es
in
a
p
o
wer
s
y
s
tem
wh
ile
s
atis
f
y
in
g
a
s
e
t
o
f
eq
u
ality
an
d
in
eq
u
ality
co
n
s
tr
ain
ts
.
Ma
th
em
atica
lly
,
th
e
OPF p
r
o
b
lem
ca
n
b
e
f
o
r
m
u
lated
as (
1
)
.
:
(
,
)
(
1
)
(
,
)
≤
0
s
u
b
ject
to
ℎ
(
,
)
≤
0
.
I
n
th
e
OPF
f
o
r
m
u
latio
n
,
r
e
p
r
esen
ts
th
e
co
n
tr
o
l
o
r
in
d
ep
e
n
d
en
t
v
ar
iab
les,
wh
ile
d
en
o
tes
th
e
s
tate
o
r
d
ep
en
d
en
t
v
ar
iab
les.
T
h
e
f
u
n
ctio
n
(
,
)
co
r
r
esp
o
n
d
s
to
th
e
o
b
jectiv
e
f
u
n
ctio
n
s
b
ein
g
o
p
tim
ized
.
T
h
e
co
n
s
tr
ain
ts
ar
e
r
e
p
r
esen
ted
as
(
,
)
f
o
r
i
n
eq
u
aliti
es
an
d
ℎ
(
,
)
f
o
r
eq
u
alities
[
6
]
-
[
8
].
2
.
1
.
Co
ntr
o
l
v
a
ri
a
bles
T
h
e
v
ar
iab
les
th
at
ac
tiv
ely
in
f
l
u
en
ce
th
e
p
o
wer
f
lo
w
with
in
t
h
e
elec
tr
ical
n
etwo
r
k
ar
e
ter
m
ed
co
n
tr
o
l
v
ar
iab
les an
d
ar
e
co
llectiv
ely
r
ep
r
esen
ted
as a
v
ec
to
r
(
2
)
.
=
[
2
…
,
1
.
.
,
1
…
,
1
…
]
(
2
)
Her
e,
r
ep
r
esen
ts
th
e
ac
t
iv
e
p
o
wer
g
en
er
ated
at
th
e
-
th
g
en
er
ato
r
b
u
s
,
ex
clu
d
in
g
th
e
s
lack
(
s
win
g
)
g
en
er
ato
r
.
Alth
o
u
g
h
b
u
s
1
is
t
y
p
ically
ch
o
s
en
as
th
e
s
win
g
b
u
s
in
s
tu
d
ies,
an
y
g
en
e
r
ato
r
b
u
s
m
ay
s
er
v
e
th
is
r
o
le.
d
en
o
tes
th
e
v
o
ltag
e
m
ag
n
itu
d
e
at
th
e
-
th
PV
(
g
en
er
ato
r
)
b
u
s
.
r
ef
er
s
to
th
e
ta
p
s
ettin
g
o
f
th
e
-
t
h
tr
an
s
f
o
r
m
er
b
r
a
n
ch
,
an
d
s
tan
d
s
f
o
r
t
h
e
s
h
u
n
t
r
ea
ctiv
e
c
o
m
p
en
s
atio
n
at
th
e
-
th
b
u
s
.
T
h
e
to
tal
n
u
m
b
e
r
o
f
g
en
er
ato
r
s
,
s
h
u
n
t
c
o
m
p
en
s
ato
r
s
,
an
d
t
r
an
s
f
o
r
m
e
r
s
is
g
iv
en
b
y
,
,
an
d
,
r
esp
ec
tiv
ely
.
E
ac
h
co
n
tr
o
l
v
ar
iab
le
is
allo
wed
t
o
v
a
r
y
wit
h
in
its
s
p
ec
if
ied
lim
its
.
W
h
ile
tr
an
s
f
o
r
m
er
tap
s
ettin
g
s
ar
e
in
h
er
en
tly
d
is
cr
ete
in
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
C
o
mp
a
r
is
o
n
o
f d
iffer
en
tia
l e
vo
lu
tio
n
o
p
timiz
a
tio
n
tech
n
iq
u
e
w
ith
o
th
er
…
(
V
in
ee
ta
S
.
C
h
a
u
h
a
n
)
665
n
atu
r
e,
th
ey
ar
e
r
e
p
r
esen
ted
h
e
r
e
in
p
er
-
u
n
it (
p
.
u
.
)
v
alu
es with
o
u
t c
o
n
s
id
er
in
g
th
e
ab
s
o
lu
te
v
o
ltag
e
m
ag
n
itu
d
e.
T
o
alig
n
with
ea
r
lier
s
tu
d
ies
an
d
f
ac
ilit
ate
co
m
p
ar
ativ
e
an
al
y
s
is
,
th
ese
tap
s
et
tin
g
s
,
as
wel
l
as
all
o
th
er
co
n
tr
o
l
v
ar
iab
les
,
ar
e
tr
ea
ted
as c
o
n
tin
u
o
u
s
f
o
r
th
e
m
ajo
r
ity
o
f
s
im
u
latio
n
s
ce
n
ar
io
s
[
6
]
-
[
8
]
.
2
.
2
.
St
a
t
e
v
a
ria
bles
T
h
e
s
tate
v
ar
iab
les
th
at
d
escr
i
b
e
th
e
p
o
wer
s
y
s
tem
s
s
tate
ar
e
r
ep
r
esen
ted
b
y
v
ec
to
r
as
(
3
)
.
Her
e
,
1
is
th
e
g
en
er
ato
r
ac
tiv
e
p
o
wer
at
s
lack
(
o
r
s
win
g
)
b
u
s
,
is
th
e
r
ea
ctiv
e
p
o
we
r
o
f
g
en
er
ato
r
co
n
n
ec
ted
to
b
u
s
,
is
th
e
b
u
s
v
o
lta
g
e
o
f
-
th
lo
ad
b
u
s
(
PQ
b
u
s
)
,
an
d
lin
e
lo
a
d
in
g
o
f
-
th
lin
e
i
s
g
iv
en
b
y
.
an
d
ar
e
th
e
n
u
m
b
er
o
f
lo
ad
b
u
s
es a
n
d
tr
an
s
m
is
s
io
n
lin
es
,
r
esp
ec
tiv
ely
[
9
]
,
[
1
0
]
.
=
[
1
,
1
…
.
,
1
…
,
1
…
]
(
3
)
2
.
3
.
Co
ns
t
ra
ints
As
m
en
tio
n
ed
b
ef
o
r
e,
b
o
th
e
q
u
ality
an
d
in
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q
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ality
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o
n
s
tr
a
in
ts
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u
s
t
b
e
s
atis
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ied
in
th
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is
s
u
e.
T
h
e
f
o
llo
win
g
lis
ts
th
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lim
it
atio
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s
.
2
.
3
.
1
.
E
qu
a
lity
co
ns
t
ra
ints
T
h
e
eq
u
ality
c
o
n
s
tr
ain
ts
in
OPF ar
e
p
o
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b
ala
n
ce
eq
u
atio
n
an
d
th
o
s
e
ar
e
ex
p
r
ess
e
d
as
(
4
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an
d
(
5
)
.
−
−
∑
=
1
[
c
os
(
)
+
s
in
(
)
]
=
0
∀
(
4
)
−
−
∑
=
1
[
s
in
(
)
+
c
os
(
)
]
=
0
∀
(
5
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W
h
er
e
=
−
,
is
th
e
d
if
f
er
en
ce
in
v
o
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n
g
les
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et
wee
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u
s
an
d
b
u
s
,
is
t
h
e
n
u
m
b
er
o
f
b
u
s
es,
an
d
ar
e
ac
tiv
e
an
d
r
ea
ctiv
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lo
ad
d
em
a
n
d
s
,
r
esp
ec
tiv
e
ly
.
is
th
e
tr
an
s
f
er
co
n
d
u
cta
n
ce
an
d
is
th
e
s
u
s
ce
p
tan
ce
b
etwe
en
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u
s
an
d
b
u
s
,
r
esp
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tiv
ely
.
2
.
3
.
2
.
I
nequ
a
lity
co
ns
t
ra
ints
I
n
th
e
OPF
f
r
am
ewo
r
k
,
in
e
q
u
ality
co
n
s
tr
ain
ts
r
ep
r
esen
t
th
e
o
p
er
atio
n
al
b
o
u
n
d
a
r
ies
o
f
s
y
s
tem
co
m
p
o
n
en
ts
,
as
well
as
th
e
lim
itatio
n
s
o
n
tr
an
s
m
is
s
io
n
lin
e
s
an
d
lo
ad
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u
s
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en
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u
r
in
g
th
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s
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u
r
e
o
p
e
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atio
n
o
f
th
e
p
o
wer
n
etwo
r
k
[
9
]
.
≤
≤
∀
(
6
)
≤
≤
∀
(
7
)
≤
≤
∀
(
8
)
C
o
n
tr
o
l
v
ar
iab
les
g
o
v
er
n
e
d
b
y
in
eq
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ality
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n
s
tr
ain
ts
ar
e
in
h
er
en
tly
r
estricte
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with
in
p
r
e
d
ef
in
ed
lim
its
.
T
h
e
o
p
tim
izatio
n
p
r
o
ce
s
s
id
en
tifie
s
s
u
itab
le
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alu
es
f
o
r
th
ese
v
a
r
iab
les
th
at
r
em
ain
with
in
th
e
ir
allo
wab
le
r
an
g
es
[
1
1
]
-
[
1
3
]
.
3.
ST
UDY
CAS
E
S AN
D
O
B
J
E
CT
I
V
E
F
UNC
T
I
O
N
S
Sev
er
al
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tu
d
y
ca
s
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in
clu
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in
g
s
in
g
le
an
d
m
u
lti
o
b
jectiv
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f
o
r
m
u
latio
n
s
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ca
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o
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t
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in
g
3
0
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u
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an
d
5
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b
u
s
test
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t
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m
s
to
ass
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th
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p
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f
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m
a
n
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d
if
f
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t
co
n
s
tr
ain
t
h
an
d
li
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g
(
C
H)
tech
n
iq
u
es.
T
ab
le
1
p
r
o
v
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es
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o
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er
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ie
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o
f
m
ain
s
y
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tem
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m
p
o
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clu
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in
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g
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r
s
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an
s
f
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m
er
s
,
a
n
d
s
h
u
n
t
co
m
p
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s
ato
r
s
,
alo
n
g
with
o
th
er
r
elev
an
t
p
ar
am
eter
s
[
1
4
]
,
[
1
5
]
.
I
n
b
o
th
s
y
s
tem
s
,
b
u
s
1
is
k
n
o
wn
as
th
e
s
win
g
(
o
r
s
lack
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b
u
s
o
r
δV
b
u
s
.
Acc
o
r
d
in
g
to
p
o
wer
b
alan
ce
(
4
)
an
d
(
5
)
,
th
e
s
win
g
b
u
s
is
es
s
en
tial
in
p
r
eser
v
in
g
s
y
s
tem
p
o
wer
b
alan
ce
a
n
aly
s
is
b
y
m
ak
in
g
u
p
f
o
r
an
y
d
is
cr
ep
an
cy
in
ac
tiv
e
an
d
r
ea
ctiv
e
p
o
wer
.
T
h
e
s
win
g
b
u
s
v
o
ltag
e
m
ag
n
itu
d
e
an
d
p
h
ase
an
g
le
a
r
e
ad
j
u
s
ted
to
1
p
.
u
.
a
n
d
0
d
eg
r
ee
s
.
As
o
u
tp
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t
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h
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d
f
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w
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aly
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is
,
th
e
v
o
ltag
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m
ag
n
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d
es
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d
p
h
ase
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g
les
o
f
ev
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u
s
ar
e
co
m
p
u
te
d
in
r
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n
to
th
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s
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g
b
u
s
.
T
h
e
n
ex
t sectio
n
s
[1
5
]
,
[1
6
]
ad
d
r
ess
th
e
cr
ea
tio
n
o
f
s
tu
d
y
ca
s
es f
o
r
th
ese
b
u
s
s
y
s
tem
s
.
3
.
1
.
P
r
o
blem
f
o
rm
ula
t
io
n
o
f
s
t
ud
y
ca
s
es
3
.
1
.
1
.
Ca
s
e
1
:
M
ini
m
iza
t
io
n
o
f
f
uel c
o
s
t
T
h
e
r
elatio
n
s
h
ip
b
etwe
en
t
h
e
f
u
el
c
o
s
t
(
in
$
/
h
)
a
n
d
t
h
e
g
en
er
ated
p
o
wer
(
in
MW)
is
ty
p
ically
m
o
d
eled
u
s
in
g
a
q
u
ad
r
atic
f
u
n
ctio
n
.
T
h
er
e
f
o
r
e,
th
e
o
b
jectiv
e
to
b
e
m
in
im
ize
d
is
ex
p
r
ess
ed
as (
9
)
.
(
,
)
=
∑
+
+
2
(9
)
W
h
er
e
,
,
ar
e
co
s
t
c
o
ef
f
icien
ts
of
th
e
-
th
g
en
er
ato
r
p
r
o
d
u
cin
g
p
o
wer
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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2
2
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T
a
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3
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4
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5
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b
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1
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R
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9
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u
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1
2
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1
5
C
o
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1
8
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2
5
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5
3
Ta
p
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r
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5
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6
1
7
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4
Tu
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3
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P
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a
n
d
(
l
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t
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d
:
2
8
3
.
4
M
W
a
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,
1
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.
2
M
V
A
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A
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4
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P
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(
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4
b
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o
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a
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d
w
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h
i
n
[
0
.
9
5
–
1
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0
5
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p
.
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.
5
0
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o
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d
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u
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mai
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t
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n
[
0
.
9
4
–
1
.
0
6
]
p
.
u
.
3
.
1
.
2
.
Ca
s
e
2
:
M
ini
m
iza
t
io
n
o
f
co
s
t
co
ns
idering
m
ulti
-
f
uels
T
o
p
r
o
d
u
ce
d
i
f
f
er
en
t
p
o
we
r
o
u
tp
u
ts
,
th
er
m
al
g
e
n
er
atin
g
p
la
n
ts
u
s
e
a
v
ar
iety
o
f
f
u
els
s
u
c
h
as
co
al,
n
atu
r
al
g
as,
an
d
o
il.
Dep
e
n
d
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n
g
o
n
th
e
k
in
d
an
d
am
o
u
n
t
o
f
f
u
el
u
s
ed
,
t
h
e
p
etr
o
l
f
u
n
ct
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n
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d
iv
id
ed
in
to
p
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wis
e
q
u
ad
r
atic
f
u
n
ctio
n
s
.
T
h
e
f
o
llo
win
g
r
ep
r
esen
ts
th
e
ex
p
en
s
e
o
f
u
s
in
g
m
a
n
y
f
u
el
s
(
1
0
)
.
(
,
)
=
+
+
(
1
0
)
W
ith
in
p
o
wer
o
u
t
p
u
t
r
a
n
g
e
≤
≤
∶
b
ein
g
th
e
f
u
el
o
p
tio
n
.
3
.
1
.
3
.
Ca
s
e
3
:
E
nh
a
ncem
ent
o
f
v
o
lt
a
g
e
s
t
a
bil
it
y
o
f
t
he
net
wo
rk
Vo
ltag
e
s
tab
ilit
y
is
s
u
es
h
av
e
g
ar
n
er
e
d
in
cr
ea
s
in
g
atten
tio
n
in
r
ec
en
t
y
ea
r
s
,
as
s
ev
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al
p
o
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s
y
s
tem
b
lack
o
u
ts
h
a
v
e
b
ee
n
attr
ib
u
te
d
to
v
o
ltag
e
in
s
tab
ilit
y
.
A
p
o
wer
s
y
s
tem
is
co
n
s
id
er
ed
v
o
lt
ag
e
s
tab
le
if
it
ca
n
r
eser
v
e
all
b
u
s
v
o
ltag
es
with
i
n
p
er
m
is
s
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le
lim
its
u
n
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er
n
o
r
m
al
o
p
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atin
g
co
n
d
itio
n
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d
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g
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tr
o
u
b
le.
Vo
ltag
e
in
s
tab
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is
es wh
en
a
tr
o
u
b
le,
in
cr
ea
s
ed
lo
a
d
d
e
m
an
d
,
o
r
a
ch
a
n
g
e
in
s
y
s
tem
c
o
n
f
ig
u
r
atio
n
lead
s
to
a
g
r
ad
u
al
an
d
u
n
co
n
tai
n
ab
le
d
ec
lin
e
in
v
o
ltag
e
lev
els
[
17
]
.
T
h
is
p
r
o
b
lem
is
p
ar
ticu
lar
ly
p
r
ev
alen
t
in
s
y
s
tem
s
ch
ar
ac
ter
ized
b
y
lo
n
g
tr
a
n
s
m
is
s
io
n
lin
es
an
d
h
ea
v
y
lo
ad
in
g
.
E
n
h
an
cin
g
v
o
ltag
e
s
tab
ilit
y
is
th
er
ef
o
r
e
a
cr
itical
asp
ec
t
o
f
p
o
wer
s
y
s
tem
o
p
er
atio
n
an
d
p
lan
n
in
g
.
T
h
e
L
-
in
d
ex
h
as
p
r
o
v
en
to
b
e
an
ef
f
ec
tiv
e
in
d
icato
r
o
f
v
o
ltag
e
s
tab
ilit
y
f
o
r
ea
ch
lo
a
d
b
u
s
in
th
e
n
etwo
r
k
[
1
8
]
.
T
h
e
in
d
e
x
r
an
g
es
f
r
o
m
0
to
1
,
wh
er
e
a
v
alu
e
o
f
0
co
r
r
esp
o
n
d
s
to
a
n
o
-
lo
ad
co
n
d
itio
n
an
d
a
v
alu
e
o
f
1
in
d
icate
s
v
o
ltag
e
co
llap
s
e.
T
h
e
L
-
in
d
ex
is
co
m
p
u
ted
f
o
r
all
lo
ad
b
u
s
es,
an
d
t
h
e
s
u
p
r
e
m
e
v
alu
e
a
m
o
n
g
th
em
is
u
s
ed
as
a
g
l
o
b
al
in
d
icato
r
o
f
th
e
s
y
s
tem
’
s
v
o
ltag
e
s
tab
ilit
y
m
ar
g
in
.
Acc
o
r
d
in
g
ly
,
th
e
o
b
jectiv
e
f
u
n
ctio
n
f
o
r
ass
ess
in
g
s
y
s
tem
s
tab
ilit
y
ca
n
b
e
d
ef
in
ed
in
ter
m
s
o
f
th
e
m
ax
im
u
m
L
-
in
d
e
x
v
alu
e
[
1
9
]
,
[
2
0
]
.
(
,
)
=
=
ma
x
(
)
,
ℎ
=
1
,
2
,
…
,
(
1
1
)
3
.
1
.
4
.
Ca
s
e
4
:
M
ini
m
iza
t
io
n
o
f
em
is
s
io
n
Pro
d
u
cin
g
elec
tr
ical
en
e
r
g
y
f
r
o
m
p
r
ed
ictab
le
f
u
el
s
o
u
r
ce
s
lead
s
to
th
e
r
elea
s
e
o
f
en
v
ir
o
n
m
en
tally
h
ar
m
f
u
l g
ases
.
T
h
e
em
is
s
io
n
lev
els
o
f
p
o
llu
ta
n
ts
s
u
ch
as
SOx
an
d
NOx
te
n
d
to
r
is
e
with
a
s
u
r
g
e
in
g
e
n
er
at
io
n
f
o
llo
win
g
th
e
f
u
n
ctio
n
al
r
elati
o
n
s
h
ip
p
r
esen
ted
in
(
1
9
)
.
T
h
er
ef
o
r
e,
r
e
d
u
cin
g
th
ese
em
is
s
io
n
s
is
co
n
s
id
er
ed
o
n
e
o
f
th
e
p
r
im
ar
y
o
b
jectiv
es in
th
e
OPF f
o
r
m
u
latio
n
.
(
,
)
=
=
∑
[
(
=
1
+
+
)
2
×
0
.
01
+
(
)
]
(
1
2
)
W
h
er
e,
,
,
,
an
d
ar
e
all
em
is
s
io
n
co
ef
f
icien
ts
f
o
r
3
0
-
b
u
s
s
y
s
tem
.
3
.
1
.
5
.
Ca
s
e
5
:
Reducing
a
ct
u
a
l po
wer
lo
s
s
B
ec
au
s
e
th
e
lin
es
in
g
ea
r
b
o
x
s
y
s
tem
s
h
av
e
in
tr
in
s
ic
r
esis
t
an
ce
,
p
o
wer
lo
s
s
is
in
ev
itab
le.
Use
th
e
f
o
llo
win
g
f
o
r
m
u
la
t
o
r
ed
u
ce
th
e
r
ea
l p
o
wer
l
o
s
s
(
in
MW)
[
2
1
]
.
(
,
)
=
=
∑
(
)
=
1
[
2
+
2
−
2
c
os
(
)
]
(
1
3
)
W
h
er
e,
=
−
,
is
th
e
d
if
f
er
en
ce
in
v
o
ltag
e
an
g
les
b
e
twee
n
b
u
s
an
d
b
u
s
,
an
d
(
)
is
th
e
tr
an
s
f
er
co
n
d
u
cta
n
ce
o
f
b
r
an
c
h
co
n
n
ec
tin
g
b
u
s
es
an
d
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
C
o
mp
a
r
is
o
n
o
f d
iffer
en
tia
l e
vo
lu
tio
n
o
p
timiz
a
tio
n
tech
n
iq
u
e
w
ith
o
th
er
…
(
V
in
ee
ta
S
.
C
h
a
u
h
a
n
)
667
3
.
1
.
6
.
Ca
s
e
6
:
Reducing
f
uel e
x
pens
es a
nd
a
ct
ua
l po
wer
l
o
s
s
T
h
e
v
o
ltag
e
q
u
ality
o
f
th
e
n
e
two
r
k
is
m
ea
s
u
r
ed
b
y
v
o
ltag
e
d
ev
iatio
n
.
Fo
r
s
ec
u
r
ity
,
th
e
d
ev
iatio
n
in
d
ex
is
cr
u
cial.
Fro
m
th
e
n
o
m
in
al
v
alu
e
o
f
u
n
ity
,
th
e
in
d
ic
ato
r
ca
lcu
lates
th
e
to
tal
ch
an
g
e
o
f
v
o
ltag
es
ac
r
o
s
s
all
lo
ad
b
u
s
es (
PQ b
u
s
es)
in
th
e
n
etwo
r
k
[
2
2
]
.
T
h
e
e
x
p
r
ess
io
n
is
as
(
1
4
)
.
(
,
)
=
∑
=
1
+
+
2
+
×
(
1
4
)
W
h
er
e
is
th
e
r
ea
l
p
o
w
er
lo
s
s
in
th
e
n
etwo
r
k
ca
lcu
la
ted
by
u
s
in
g
(
1
3
)
.
3
.
1
.
7
.
Ca
s
e
7
:
M
ini
m
iza
t
io
n
o
f
f
uel c
o
s
t
a
nd
v
o
l
t
a
g
e
dev
i
a
t
io
n
T
h
e
v
o
ltag
e
q
u
ality
o
f
th
e
n
e
two
r
k
is
m
ea
s
u
r
ed
b
y
v
o
ltag
e
d
ev
iatio
n
.
Fo
r
s
ec
u
r
ity
,
th
e
d
ev
iatio
n
in
d
ex
is
cr
u
cial.
F
r
o
m
th
e
n
o
m
in
al
v
alu
e
o
f
u
n
ity
,
th
e
in
d
ic
ato
r
ca
lcu
lates
th
e
to
tal
ch
an
g
e
o
f
v
o
ltag
es
ac
r
o
s
s
all
lo
ad
b
u
s
es (
PQ b
u
s
es)
in
th
e
n
etwo
r
k
[
2
3
]
.
T
h
e
e
x
p
r
ess
io
n
is
(
1
5
)
.
=
(
∑
|
=
1
−
1
|
)
(1
5
)
T
h
e
co
m
b
i
n
ed
o
b
jectiv
e
f
u
n
ctio
n
o
f
f
u
el
c
o
s
t a
n
d
v
o
ltag
e
d
e
v
iatio
n
is
(
1
6
)
.
(
,
)
=
(
∑
+
+
2
)
+
×
−
1
(1
6
)
3
.
1
.
8
.
Ca
s
e
8
:
I
m
pro
v
ing
v
o
l
t
a
g
e
s
t
a
bil
it
y
a
nd
re
du
cing
f
uel c
o
s
t
T
h
is
o
b
jectiv
e
f
u
n
ctio
n
s
ee
k
s
to
im
p
r
o
v
e
s
y
s
tem
v
o
ltag
e
s
tab
ilit
y
an
d
lo
wer
f
u
el
ex
p
e
n
s
es.
A
s
in
g
le
aim
is
cr
ea
ted
b
y
c
o
m
b
in
in
g
s
ev
er
al
o
b
jectiv
es
as
(
1
7
)
.
(
,
)
=
(
∑
+
+
2
)
+
×
−
1
(
1
7
)
W
h
er
e
is
ca
lcu
lated
u
s
i
n
g
(
1
5
)
.
3
.
1
.
9
.
Ca
s
e
9
:
M
ini
m
iza
t
io
n
o
f
f
uel c
o
s
t
,
e
m
is
s
io
n,
v
o
lt
a
g
e
dev
ia
t
io
n
,
a
nd
lo
s
s
es
R
ed
u
cin
g
f
u
el
co
s
t,
lo
wer
in
g
p
o
llu
tan
ts
,
m
in
im
izin
g
v
o
ltag
e
d
ev
iatio
n
s
,
an
d
d
ec
r
ea
s
in
g
r
ea
l
p
o
wer
lo
s
s
es
ar
e
th
e
f
o
u
r
m
ain
g
o
als
th
at
th
is
ca
s
e
s
tu
d
y
s
im
u
ltan
eo
u
s
ly
tack
les.
T
h
e
f
o
llo
win
g
i
s
th
e
f
o
r
m
u
latio
n
o
f
co
m
b
in
ed
o
b
jectiv
e
f
u
n
ctio
n
(
1
8
)
.
(
,
)
=
(
∑
+
+
2
)
+
×
+
×
+
+
(
1
8
)
T
o
b
alan
ce
th
e
o
b
jectiv
es,
th
e
weig
h
t f
ac
to
r
s
ar
e
s
elec
ted
wit
h
=
19,
=
21
,
a
n
d
=
22
.
3
.
2
.
P
er
f
o
r
m
a
nce
ev
a
lua
t
io
n o
n IE
E
E
5
7
-
b
us
t
est
s
y
s
t
em
T
h
e
ess
en
tial
co
n
f
ig
u
r
atio
n
a
n
d
p
ar
am
eter
v
alu
es
o
f
th
e
I
E
E
E
5
7
-
b
u
s
test
s
y
s
tem
ar
e
o
u
tlin
ed
in
T
ab
le
1
.
T
o
e
v
alu
ate
alg
o
r
ith
m
p
er
f
o
r
m
an
ce
u
n
d
e
r
v
ar
io
u
s
o
p
tim
izatio
n
co
n
d
itio
n
s
,
f
o
u
r
d
is
tin
ct
ca
s
e
s
tu
d
ies
ar
e
co
n
d
u
cted
.
T
h
ese
in
clu
d
e
two
s
in
g
le
-
o
b
jectiv
e
an
d
two
m
u
lti
-
o
b
jectiv
e
o
p
tim
izatio
n
p
r
o
b
lem
s
,
en
s
u
r
in
g
a
b
alan
ce
d
ex
p
lo
r
atio
n
o
f
al
g
o
r
i
th
m
ef
f
ec
tiv
en
ess
.
3
.
2
.
1
.
Ca
s
e
1
0
:
Reducing
f
uel c
o
s
t
T
h
is
ca
s
e
ad
d
r
ess
es
a
s
in
g
le
-
o
b
jectiv
e
o
p
tim
al
p
o
wer
f
lo
w
(
OPF
)
p
r
o
b
lem
f
o
c
u
s
ed
o
n
m
in
im
izin
g
th
e
to
tal
g
en
er
atio
n
f
u
el
c
o
s
t.
T
h
e
m
ath
e
m
atica
l
f
o
r
m
u
latio
n
o
f
th
e
co
s
t
f
u
n
ctio
n
a
n
d
ass
o
ciate
d
co
e
f
f
icien
ts
(
f
u
el
c
o
s
t
an
d
em
is
s
io
n
p
ar
a
m
eter
s
)
r
em
ain
co
n
s
is
ten
t
with
th
o
s
e
e
m
p
lo
y
e
d
in
th
e
I
E
E
E
3
0
-
b
u
s
b
en
ch
m
a
r
k
s
y
s
tem
,
as d
escr
ib
ed
in
[
2
4
]
,
[
2
5
]
.
3
.
2
.
2
.
Ca
s
e
1
1
:
Co
m
bin
ed
re
du
ct
io
n o
f
f
uel c
o
s
t
a
nd
v
o
lt
a
g
e
dev
ia
t
io
n
I
n
th
is
m
u
lti
-
o
b
jectiv
e
s
ce
n
ar
i
o
,
th
e
o
b
jectiv
e
is
to
s
im
u
ltan
eo
u
s
ly
r
ed
u
ce
t
h
e
f
u
el
c
o
s
t
an
d
th
e
to
tal
v
o
ltag
e
d
e
v
iatio
n
ac
r
o
s
s
th
e
l
o
ad
b
u
s
es.
T
h
ese
two
g
o
als
a
r
e
co
m
b
in
ed
i
n
to
a
s
ca
lar
o
b
j
ec
tiv
e
f
u
n
ctio
n
b
y
ap
p
ly
in
g
a
weig
h
t
f
ac
to
r
ass
ig
n
ed
a
v
al
u
e
o
f
1
0
0
.
T
h
is
s
ca
lar
izatio
n
m
eth
o
d
m
ir
r
o
r
s
th
e
ap
p
r
o
ac
h
u
s
ed
p
r
ev
io
u
s
ly
i
n
th
e
I
E
E
E
3
0
-
b
u
s
s
y
s
tem
to
h
an
d
le
c
o
n
f
lictin
g
o
b
jectiv
es.
3
.
2
.
3
.
Ca
s
e
1
2
:
Co
m
bin
ed
re
du
ct
io
n o
f
f
uel c
o
s
t
m
ini
m
iz
a
t
io
n a
nd
v
o
lt
a
g
e
s
t
a
bil
it
y
en
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el
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u
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tifie
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en
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tr
a
d
e
-
o
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f
s
in
o
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tim
izatio
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
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:
2
2
5
2
-
8
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2
I
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2
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3
.
2
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4
.
Ca
s
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1
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Reducing
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p
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ally
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p
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th
e
I
E
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b
u
s
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tem
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en
s
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g
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eth
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o
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ile
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ap
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s
ca
le
o
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e
5
7
-
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u
s
n
etwo
r
k
.
4.
AL
G
O
RI
T
H
M
F
O
R
DIFF
E
RE
N
T
I
A
L
E
VO
L
U
T
I
O
N
(
D
E
)
AL
G
O
RI
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H
M
T
h
e
r
esil
ien
t,
s
to
ch
asti
c
p
o
p
u
latio
n
-
b
ased
o
p
tim
izatio
n
m
eth
o
d
k
n
o
wn
as
DE
,
wh
ich
was
f
ir
s
t
in
tr
o
d
u
ce
d
b
y
[
2
6
]
,
[
2
7
]
,
h
as
b
ec
o
m
e
p
o
p
u
lar
f
o
r
s
o
lv
in
g
c
o
m
p
licated
an
d
n
o
n
lin
ea
r
o
p
tim
izatio
n
p
r
o
b
lem
s
.
A
p
o
p
u
latio
n
o
f
p
o
te
n
tial
s
o
lu
tio
n
s
is
iter
ativ
ely
r
ef
in
ed
b
y
th
e
alg
o
r
ith
m
th
r
o
u
g
h
a
cy
cle
o
f
m
u
tatio
n
,
cr
o
s
s
o
v
er
,
an
d
s
elec
tio
n
.
A
f
it
n
ess
f
u
n
ctio
n
is
u
s
ed
to
ass
ess
ea
ch
m
em
b
er
o
f
th
e
p
o
p
u
latio
n
,
an
d
th
o
s
e
th
at
d
o
b
etter
ar
e
r
etain
ed
to
af
f
ec
t
th
e
m
a
k
eu
p
o
f
th
e
f
o
llo
win
g
g
en
er
atio
n
.
An
o
v
er
v
iew
o
f
th
e
m
ain
s
tep
s
in
th
e
DE
p
r
o
ce
s
s
is
g
iv
en
in
th
is
s
ec
tio
n
.
4
.
1
.
I
nitia
liza
t
io
n o
f
po
pu
la
t
io
n
E
ac
h
ca
n
d
id
ate
s
o
lu
ti
o
n
(
o
r
d
ec
is
io
n
v
ec
to
r
)
is
g
iv
e
n
a
r
an
d
o
m
v
alu
e
with
in
t
h
e
s
p
ec
if
ied
b
o
u
n
d
s
o
f
th
e
co
r
r
esp
o
n
d
in
g
d
ec
is
io
n
v
a
r
iab
les.
At
th
e
s
tar
t
o
f
th
e
DE
p
r
o
ce
s
s
,
th
at
s
tar
ts
with
th
e
r
a
n
d
o
m
g
e
n
er
atio
n
o
f
an
in
itial
p
o
p
u
latio
n
.
T
h
e
in
iti
aliza
tio
n
g
u
ar
an
tees
th
at
th
e
s
ea
r
ch
b
eg
in
s
in
th
e
s
o
l
u
tio
n
s
p
ac
e
v
iab
le
r
eg
i
o
n
.
T
h
e
s
tar
tin
g
v
alu
es o
f
th
e
k
t
h
d
ec
is
io
n
v
ec
to
r
a
n
d
jth
c
o
m
p
o
n
en
t a
r
e
as
(
1
9
)
.
.
0
(
)
=
(
)
+
,
[
0
,
1
]
×
[
m
ax
(
)
−
(
)
]
(
1
9
)
W
h
er
e
,
[
0
,
1
]
is
a
r
an
d
o
m
n
u
m
b
er
ly
i
n
g
b
etwe
en
0
a
n
d
1
;
=
1
,
2
,
…,
wh
er
e
is
th
e
d
im
en
s
io
n
o
f
th
e
d
ec
is
io
n
v
ec
to
r
.
4.
2
.
M
uta
t
io
n
I
n
th
e
s
ec
o
n
d
s
tep
o
f
th
e
DE
alg
o
r
ith
m
,
a
m
u
tatio
n
o
p
e
r
atio
n
is
ap
p
lie
d
to
g
en
e
r
ate
a
m
u
tan
t
(
o
r
d
o
n
o
r
)
v
ec
to
r
Vk
,
G
f
o
r
ea
ch
in
d
iv
id
u
al
in
t
h
e
p
o
p
u
latio
n
,
r
ef
er
r
ed
t
o
as
th
e
tar
g
et
v
ec
to
r
Xk
,
G
wh
er
e
th
e
s
u
b
s
cr
ip
t
‘
G’
d
en
o
tes
th
e
cu
r
r
en
t
g
e
n
er
atio
n
.
T
h
e
m
u
tati
o
n
s
tr
ateg
y
em
p
lo
y
ed
in
th
is
wo
r
k
f
o
llo
ws
th
e
“DE
/r
an
d
/1
”
s
ch
em
e,
wh
e
r
e
th
e
m
u
tan
t
v
ec
to
r
is
f
o
r
m
ed
b
y
ad
d
in
g
th
e
weig
h
ted
d
if
f
er
en
ce
o
f
two
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an
d
o
m
l
y
s
elec
ted
p
o
p
u
latio
n
v
ec
to
r
s
to
a
th
ir
d
r
a
n
d
o
m
l
y
ch
o
s
en
v
ec
to
r
(
2
0
)
.
,
=
1
,
+
(
2
,
−
3
,
(
2
0
)
T
h
e
in
d
ices
1
,
2
,
a
n
d
3
ar
e
d
is
tin
ct
an
d
r
a
n
d
o
m
l
y
s
elec
ted
f
r
o
m
th
e
en
tire
p
o
p
u
latio
n
r
an
g
e.
T
h
e
m
u
tatio
n
f
ac
to
r
is
a
p
o
s
itiv
e
p
ar
am
eter
u
s
ed
t
o
s
ca
le
th
e
d
if
f
er
en
ce
b
etwe
en
v
ec
to
r
s
.
4
.
3
.
Cro
s
s
o
v
er
T
h
e
tr
ial
o
r
o
f
f
s
p
r
in
g
v
ec
to
r
,
=
[
,
(
1
)
,
,
(
2
)
,
.
.
.
,
,
(
)
]
is
g
en
er
ated
b
y
jo
in
in
g
elem
en
ts
o
f
th
e
d
o
n
o
r
v
ec
to
r
with
th
o
s
e
o
f
th
e
tar
g
et
v
ec
to
r
,
th
r
o
u
g
h
a
cr
o
s
s
o
v
er
p
r
o
ce
s
s
.
I
n
th
is
ap
p
r
o
ac
h
,
b
in
o
m
ial
cr
o
s
s
o
v
er
is
em
p
lo
y
ed
,
wh
er
e
ea
ch
c
o
m
p
o
n
en
t
is
n
o
m
in
ated
b
ase
d
o
n
a
co
m
p
ar
is
o
n
b
etwe
en
a
r
a
n
d
o
m
ly
g
e
n
er
at
ed
n
u
m
b
er
(
r
an
g
in
g
f
r
o
m
0
to
1
)
an
d
a
p
r
e
d
ef
in
e
d
cr
o
s
s
o
v
er
r
ate
.
T
h
e
p
r
o
ce
d
u
r
e
f
o
r
d
eter
m
in
in
g
ea
c
h
elem
en
t is o
u
tlin
ed
as
(
2
1
)
.
,
(
)
=
{
,
(
)
ℎ
,
ℎ
=
1
,
…
,
,
(
)
=
,
[
0
,
1
]
≤
(
2
1
)
W
h
er
e,
is
a
r
an
d
o
m
ly
c
h
o
s
en
n
at
u
r
al
n
u
m
b
er
i
n
{1
,
2
,
…,
D}.
T
h
e
o
f
f
s
p
r
in
g
v
ec
t
o
r
Uk
,
GU
{k
,
G}
Uk
,
G
is
ev
alu
ated
an
d
co
m
p
ar
e
d
with
th
e
p
ar
e
n
t
v
ec
to
r
Xk
,
GX
{k
,
G}
Xk
,
G
b
ased
o
n
f
itn
ess
an
d
co
n
s
tr
ain
t
v
io
latio
n
.
I
n
co
m
p
u
tatio
n
ally
in
ten
s
iv
e
r
ea
l
-
wo
r
ld
p
r
o
b
lem
s
lik
e
OPF,
tr
ial
-
an
d
-
er
r
o
r
to
id
en
tif
y
a
s
u
itab
le
C
H
m
eth
o
d
ca
n
b
e
in
ef
f
icien
t.
T
o
a
d
d
r
ess
th
is
,
th
e
E
C
HT
m
eth
o
d
co
m
b
in
es
m
u
lt
ip
le
C
H
tech
n
iq
u
es
(
SF
an
d
SP
)
with
DE
[
1
9
]
as
th
e
b
ase
o
p
tim
izatio
n
alg
o
r
ith
m
.
E
ac
h
C
H
tech
n
iq
u
e
m
ain
tain
s
its
o
wn
p
o
p
u
latio
n
an
d
p
ar
a
m
eter
s
,
p
r
o
d
u
cin
g
o
f
f
s
p
r
in
g
th
at
n
o
t
o
n
l
y
co
m
p
ete
with
in
th
eir
o
wn
g
r
o
u
p
b
u
t
also
ac
r
o
s
s
th
e
p
o
p
u
latio
n
s
o
f
o
th
e
r
C
H
m
eth
o
d
s
.
T
h
is
cr
o
s
s
-
co
m
p
etit
io
n
en
s
u
r
es
th
at
ev
en
if
a
n
o
f
f
s
p
r
in
g
is
r
ejec
ted
with
in
its
o
wn
g
r
o
u
p
,
it
m
a
y
s
till
b
e
ac
ce
p
ted
elsewh
er
e
—
th
u
s
m
ax
im
izin
g
th
e
u
tili
ty
o
f
ea
ch
f
u
n
ctio
n
ev
alu
atio
n
.
E
C
HT
d
y
n
am
icall
y
ad
ap
ts
,
allo
win
g
th
e
m
o
s
t
e
f
f
ec
tiv
e
C
H
tech
n
iq
u
e
at
a
n
y
s
tag
e
to
d
o
m
i
n
ate
an
d
in
f
lu
e
n
ce
th
e
ev
o
lu
tio
n
ar
y
p
r
o
ce
s
s
.
T
h
is
r
em
o
v
es
th
e
n
ee
d
f
o
r
m
a
n
u
al
tu
n
i
n
g
an
d
m
e
th
o
d
s
elec
tio
n
.
Fo
r
th
e
OPF
p
r
o
b
lem
,
th
e
d
ec
is
io
n
v
ar
iab
le
d
im
en
s
io
n
s
(
)
ar
e
2
4
,
3
3
,
a
n
d
1
3
0
f
o
r
t
h
e
3
0
-
a
n
d
5
7
-
b
u
s
s
y
s
tem
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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p
l Po
wer
E
n
g
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SS
N:
2252
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8
7
9
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C
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a
r
is
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f d
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tia
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lu
tio
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w
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(
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669
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Fig
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n
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m
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d
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s
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o
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tim
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(
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ases
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r
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r
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ase
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m
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r
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ases
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ates
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ase
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ical
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ts
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wh
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m
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u
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